This was going to be a comment to Differentiable structures on R^3, but I thought it would be better asked as a separate question.
So, it's mentioned in the previous question that $\mathbb{R}^4$ has uncountably many (smooth) differentiable structures. This is a claim I've certainly heard before...

you can just call it graduate level. In terms of content or material, there isn't much actual difference between a masters and a PhD. Masters degrees cover most if not all of the actual coursework/learning in graduate school, the distinguishing feature for a PhD is mostly the added work they put into it: publications, etc. Obviously that may vary from institution to institution, but it's the general trend I've noticed

@yuggib How does the notion of "individual subsystems" disappear as I progressively made two qubits more entangled from 0 to 100%., because from the density matrix of the full system, all I see is the diagonal entries of the matrix goes from some fractions that adds to 1 to becoming 1/2 for both, and that when > 0% entangled, I cannot split them into subsystems via tensor products?

@Jim yeah, without wanting to hurt your feelings, from my side of the atlantic it appeared like the edutcation system is pretty similar even if I'm aware that it's not the same country - just interesting to hear that there's no big difference between Master & PhD

Is there some secret PhD society where they have hidden physics that nobody speaks about around non-PhD people. And when studying for a PhD, you take courses where they are like "Okay, here is the REAL science that we keep secret because everyone else is stupid"

@Sanya Coursework learning, there's little difference. That means, except in the field of one's research, a masters and a PhD have about the same level of understanding of physics in general. But in the field of research, masters grad <<< PhD grad

Wacken Open Air (W:O:A) is a summer open-air heavy metal music festival. It takes place annually in the small village of Wacken in Schleswig-Holstein, northern Germany. With 80,000 festival visitors, and including personnel a total of roughly 86,000 attendees in 2011, it attracts various kinds of metal and hard rock music fans.
The festival was first held in 1990 as a small event for local German bands. W:O:A is usually held at the beginning of August and lasts now four days. It is currently considered the biggest heavy metal festival in the world.
The festival ends traditionally on the first Sunday...

it basically depends on your learning; most of the questions here are lazy questions in a sense that given a good library, research skills and time they are absolutely doable by yourself - do you want to answer it? then go ahead :D

Well, I met with a physics post doc last week to share a set of original research I have been working on. I was surprised to find that he thinks I shouldn't try for a Ph.D. and should, instead collate my ideas into a 4-10 page research paper.

he thinks that obtaining a PhD would be impractical because the theoretical stuff I am interested in is not currently "in vogue," and that I would be forced to do research in things I don't care about, and also that the skills I would gain doing a PhD aren't worth the effort

Yeah, I know. I should really be working, myself

I come here mostly to lurk, and absorb physics talk from people who likely know better than I do

@DanielSank Fair enough. I'm still standing by my opinion that de-capitalizing the title is not enough to count as an actual improvement on the post. It can accompany something that actually improves the post, but I'd still reject as "no significant improvement"

the larger issue, however, was that once it was rejected, the suggester immediately suggested what was essentially the exact same edit. Ignoring the opinions of the reviewers (plural, I wasn't the only one)

@DanielSank seriously? I read the title perfectly fine both with and without the edit. It's true that consistency is ideal, but it should not itself be a reason to bump something to the top of the active page. Especially when it is ensuring such a minor consistency correction

I, on the other hand, think that changing the first letter of half of the words in the title to be in lower case does not improve the readability of the post enough to be worth its own edit. If they had added even one tag, I'd have approved it.

@WilliamBulmer it seems surprising to hear a Phd denigrate the achievement. what does he think is "in vogue"? what is your research on? do you have a masters? we are always looking for guest speakers in here

@DanielSank if the crap improves readability, I see no problem with allowing it. I feel what I more going to do is keep someone from making repeated minor edits to a question so that their answer can stay active longer and get more views. And as a robot, I feel making a judgement call about intentions in the moment is a bit beyond what should be expected of me

@vzn I thought it was odd, too. He didn't really elaborate on what was currently in vogue, so unfortunately, I can't answer that question. My research concerns Classical Electrodynamics. No, I don't have a Master's. I have 3 years formal college -training plus a few more years of private study. I am currently making my way through Rindler's GR and am trying to learn about Lie Theory, as it seems to be fairly useful to know about.

He thinks what I really need, more than anything else, is feedback on my ideas

Also, I don't want to make it sound like he endorses my ideas (yet). He just thinks it would be better for me to try to publish a paper on them, should they be sound, than for me to try obtaining a PhD

Actually, I do have a question, which might help me. Does anyone know what the significance of the eigenvalues of principal stresses of a stress tensor is? I suppose it is likely related to treating the stress tensor as an operator, but I don't quite understand what it would be operating on. Would it be operating on field velocity?--to obtain a certain kind of field momentum density?

These don't seem very bad to me. If we compare this to the previous decades, I don't think it had been significantly better or worse, although I admit my naive impression based on remembers from the news is probably below 5 sigma. :-)

Well, and the WMAP + 1a supernovas + some third experiment, I forgot which... these 3 independent measurement gave an consequent and accurate answer to the age and the curvature of the Universe.

Although maybe the flat geometry was a little bit sad. Probably most of us waited a small positive curvature. Everything is spherical in the Universe, already Aristotle said, that the planets are spherical, and the sky is spherical, because the sphere is the most perfect geometrical form. :-)

@ACuriousMind Given the flow $\varphi_t$ of a vector field $X$ on a compact manifold $M$, $\varphi_t:M\to M$ for fixed $t$ is a diffeomorphism. Is there a general formula for $(\mathrm d\varphi_t)_p(v)$ for $p\in M, v\in T_pM$?

The Maxwell stress tensor (named after James Clerk Maxwell) is a second rank tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor...

In continuum mechanics, the Cauchy stress tensor
σ
{\displaystyle {\boldsymbol {\sigma }}\,\!}
, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components
σ
i
j
{\displaystyle \sigma _{ij}\,\!}
that completely define the state of stress at a point inside a material in the deformed state, placement...

@WilliamBulmer That's not some deep physics, it's just linear algebra: Being an eigenvalue of $T$ means that $\det(T - \lambda I)=0$. The Maxwell stress is (modulo constants) $T = E\otimes E+B\otimes B + \rho I$ where $\rho$ is the density. The rank of $E\otimes E$ and $B\otimes B$ is 1 (or 0), so $T-\rho I$ has maxmal rank 2, and as a 3x3 matrix it will therefore have $\rho$ as an eigenvalue.

@WilliamBulmer : the way the electromagnetic stress tensor resembles the continuum-mechanics Cauchy stress tensor. And the way the expression for the speed of a shear wave in a solid c = √(G/ρ) resembles c = 1/√(ε0μ0). The reciprocal is there because permittivity is a "how-easy" measure rather than a "how-hard" measure.

I'm working through the scattering sections of Mechanics by Landau and Lifshitz, and wanted to know if/how physicists today employ the methods of purely classical scattering processes. As far as I can surmise, these are useful for
Understanding Rutherford Scattering in the gold foil experiment ...

@ACuriousMind Let $(S^2,g)$ be a gradient Ricci soliton. One can show that that soliton function $f:S^2\to \Bbb R$ satisfies $\nabla^2f=(\rho-\frac{1}{2}s)g$, where $s$ is the scalar curvature and $\rho$ is the soliton constant. Since $S^2$ is 2D, one can show (cf. Kobayashi-Nomizu Vol 2) that there is an almost complex structure $J$ and a symplectic form $\omega$ such that $g(u,Jv)=\omega(u,v)$ and $\omega(u,Jv)=g(u,v)$. Furthermore, $\nabla J=0$. Letting $\xi=\mathrm{grad}(f)$, one can [cont.]

@ACuriousMind show that $J\xi$'s flow is a 1-parameter group of isometries. Let $p\in S^2$ be a critical point of $p$. Let $a=\rho-\frac{1}{2}s(a)$. Then $D^2f=ag$ at $p$. SOMEHOW this implies $(\mathrm d\varphi_t)_p(v)=\cos(at)v+\sin(at)Jv$ for all $v\in T_pS^2$.

@ACuriousMind I don't think Ricci flow is important here

One uses Ricci solitons to construct Ricci flows, a priori they are not linked

@ACuriousMind Still, we have the two other eigenvalues to account for, and they, too have units in energy density

@ACuriousMind But there is significance to its eigenvectors, which are the principal stresses

@ACuriousMind And surely, one could come up with vectors for this to act on, right? I mean, the canonical example would be faces that experience no "shear" transfers of momentum (I don't say forces, because the Maxwell stress tensor seems to represent a generalized momentum flow, rather than an actual force like the Lorentz force).

@WilliamBulmer I'm not an expert on elasicity theory, but I think the stresses are the eigenvalues, not the eigenvectors. Also, the principal stresses - like the principal axes of the moment of inertia - are nice because of computational simplicity. You can compute everything you can compute from them in any other basis, it's just going to be more tedious.

@WilliamBulmer What do you mean, "surely, one could come up with vectors for this to act on"?

@ACuriousMind If $\sigma$ represented a "force density", which it doesn't, it would represent the forces on a volume element of material. However, it is more of a flow (as I understand it. Maybe I am wrong)

@ACuriousMind hold on...I should have finished my thought before writing that

Well, okay - you let it act on normal vectors to get stress vectors on the planes they're normal to. Using the eigenvectors gives you the principal stresses, which are just the same normal vector multiplied by the eigenvalue.

@WilliamBulmer Since it's the spatial part of the EM stress-energy, it represents a pressure (diagonal elements) and shear stresses (off-diagonals).

So yes, it does represent a force density - pressure is force per unit area after all.

@ACuriousMind So, let's approach this from a different angle. Yes, I get that it represents pressure. On the other hand, it also has units in energy density. To me, that suggests that there is an interpretation involving energy density